
We discuss how \(F\)  is computed in sec:Speed. To solve (REF ), we approximate the spatial derivatives using a second order essentially non-oscillatory scheme (see [1]} and [2]} for details), and integrate in time using second order total variation diminishing (TVD) Runge-Kutta (RK), which is performed by taking two forward Euler steps
\(\tilde{\phi }^{(n + 1)} &= \phi ^{(n)} - \Delta t F^{(n)} | \mbox{$\nabla $}\phi ^{(n)} |,\\\phi ^{(n + 2)} &= \tilde{\phi }^{(n+1)} - \Delta t F^{(n+1)} | \mbox{$\nabla $}\tilde{\phi }^{(n+1)} |,\) 
